The squeezed dark nuclear spin state in lead halide perovskites

Coherent many-body states are highly promising for robust quantum information processing. While far-reaching theoretical predictions have been made for various implementations, direct experimental evidence of their appealing properties can be challenging. Here, we demonstrate optical manipulation of the nuclear spin ensemble in the lead halide perovskite semiconductor FAPbBr3 (FA = formamidinium), targeting a long-postulated collective dark state that is insensitive to optical pumping after its build-up. Via optical orientation of localized hole spins we drive the nuclear many-body system into this entangled state, requiring a weak magnetic field of only a few milli-Tesla strength at cryogenic temperatures. During its fast establishment, the nuclear polarization along the optical axis remains small, while the transverse nuclear spin fluctuations are strongly reduced, corresponding to spin squeezing as evidenced by a strong violation of the generalized nuclear squeezing-inequality with ξs < 0.5. The dark state corresponds to an ~35-body entanglement between the nuclei. Dark nuclear spin states can be exploited to store quantum information benefiting from their long-lived many-body coherence and to perform quantum measurements with a precision beyond the standard limit.

A photo of the hybrid organic-inorganic FAPbBr 3 perovskite crystal under study is shown in the inset of Supplementary Fig. 1(a).It is a crystal with a size of 5 × 5 × 2 mm 3 and a glossy semi-transparent reddish color.Its low temperature photoluminescence measured at T = 5 K has maximum at 2.1776 eV, see Supplementary Fig. 1(a).
In the time-resolved Kerr ellipticity (TRKE) measurements the laser was tuned to the energy of 2.191 eV, resonant with the free exciton.The pump beam was helicity modulated between σ + /σ − circular polarization at the frequency of 50 kHz to avoid dynamical nuclear polarization.A typical TRKE dynamics trace measured in B V = 0.5 T magnetic field applied in the Voigt geometry is shown in Supplementary Fig. 1(b).It is contributed by the coherent spin dynamics of charge carriers, which spins are initially oriented by the circularly polarized pump and then precess about the magnetic field.One can clearly see two Larmor precession frequencies, corresponding to electrons (e) and holes (h), which is the typical phenomenology for lead halide perovskite crystals [S1].We have shown that the coherent spin dynamics in perovskite semiconductors are provided by resident electrons and holes, which are localized in spatially different sites [S2-S4].
The TRKE dynamics were fitted using the function: T RKE = A e cos (Ω L,e t) exp (−t/T * 2,e ) + A h cos (Ω L t) exp (−t/T * 2,h ), (1) as shown by the yellow dashed line in Supplementary Fig. 1(b).A e(h) are the Kerr ellipticity amplitudes, reflecting the degrees of carrier spin polarization, Ω L = g h µ B B/ and Ω L,e = g e µ B B/ are the Larmor frequencies with the carrier g-factors g e(h) , and T * 2,e(h) are the ensemble spin dephasing times.There we show the individual components of the hole (blue line) and the electron (red line) spin dynamics obtained form the fit with the parameters g h = +0.4 and T * 2,h = 780 ps for the holes and g e = +2.4 and T * 2,e = 1080 ps for the electrons.The assignment of the components is made based on the universal dependence of their g-factors on the band gap energy that we have published recently [S1].The time delay is fixed at the negative value of −75 ps, marked by the arrow in Supplementary Fig. 1(b), for the measurements of the resonant spin amplification (RSA) and polarization recovery curves (PRCs).

B. Nuclear spin polarization in tilted magnetic field
Using constant pump helicity, additional measurements of the nuclear spin polarization were performed using the same scheme as for Fig. 1.Supplementary Figures 2(a,b) show that the Overhauser field for the electrons does not depend on the strength of the external magnetic field, as expected for dynamic nuclear  spin polarization [S5].The Overhauser field for the holes decreases from 22 mT at B = 150 mT to 12 mT at B = 250 mT and then stays about constant at 10 mT up to the magnetic field of 750 mT.Nevertheless, it remains small in comparison with its typical stochastic fluctuation in equilibrium, ∆ B = 40 mT.
In the angular dependence of the Overhauser field for both electron and hole spins, despite a certain scattering of the achieved field value, a dependence according to a cosine-function is found, see Supplementary Figs.2(c,d).The angular dependence of the Overhauser field experienced by the holes for several magnetic fields is given in Supplementary Fig. 2(e).

C. PRC at constant helicity pumping
In order to analyze the limit of f mod → 0, we performed additional PRC measurements for constant pump helicity, σ + or σ − .Supplementary Figure 3(a) shows the two corresponding Kerr ellipticity dynamics.They merge with each other when reversing amplitude and magnetic field.In particular, when their amplitudes are summed, the narrow peak around B F = 0 becomes fully compensated and cannot be seen [Supplementary Fig. 3(b)].We plot also the difference of the signals, which corresponds to the nuclear spin inertia signal at f mod = 0 and compare it with the case of f mod = 20 Hz in Supplementary Fig. 3(c).Generally, the signals are similar, however the latter shows two weak maxima at B F ≈ ±10 mT around the zero-field dip in agreement with the theoretical modeling, see the green curve in Fig. 3(f).

D. PRC power dependence
Here we present data for the pump power dependence of the PRC signals.In Supplementary Figures 4(a,b) we compare the PRC curves for two different regimes: when the DNSS is formed (f mod = 20 Hz) and in absence of the DNSS (f mod = 5 kHz).
All curves were evaluated by fits with respect to their amplitude, the ratio of broad dip (A b ) to the high field (A hf ) amplitude and the dip width, shown in Supplementary Figs. 4(b,c,d), respectively.To start with 5 kHz modulation frequency (full symbols), with increasing pump power the amplitude of the PRC dip and the high field amplitude rise linearly before showing a saturation trend.The narrow dip shows up as a minor peak with magnitude and width near the noise level so that is cannot be evaluated reliably for this frequency.The ratio of the PRC dip to the amplitude at high magnetic fields (≈ 150 mT) [Supplementary Fig. 4(c)] decreases slightly with increasing pump power.The width stays about constant in Supplementary Fig. 4(d) for 5 kHz modulation.
In contrast, for the low modulation frequency of 20 Hz, the PRC shape changes drastically with varying pump power.Again, the amplitudes at high field (A hf ), of the broad dip (A b ) and of the narrow dip (A n ) rise nonlinearly with a tendency to saturate with increasing pump power.However, by contrast to the previous case, the ratio A b /A hf here drops in a nonlinear fashion much stronger.Further, the width of the broad peak broadens significantly with excitation power.The width of the narrow peak remains constant.Note, in both cases of 5 kHz and 20 Hz, the power dependence of the spin signals match each other at the high magnetic field of B F = ±150 mT, when the hyperfine interaction plays a minor role, see the blue symbols.
Overall the PRC pump power dependence for the low modulation frequency [Supplementary Fig. 4(a)] is similar to the frequency dependence, shown in Fig. 2(b).As the DNSS formation rate is determined by the flux of angular momentum to the nuclear spin system, this shows that it is also proportional to the flux of angular momentum from exciting photons to a hole in agreement with the theoretical prediction, Supplementary Eq. ( 20).
E. DNSS at zero magnetic field Surprisingly, we do not observe the hole spin polarization recovery with decreasing modulation frequency at zero magnetic field, which evidences the fragility of the DNSS in the field range 1 mT.The exact reason for the DNSS not to form at zero field is unclear so far, but we note that the ODNMR in Fig. 4 has the same linewidth of about 1 mT as the narrow PRC dip.This suggests that the DNSS is destroyed by the nuclear dipole-dipole interactions either between Pb nuclei or with other nuclei such as Br.The weak dependence of the HWHM of this component on the modulation frequency shown in Fig. 2(c) supports this suggestion.Application of the magnetic field with strengths exceeding that of the local fields suppresses this interaction and stabilizes the DNSS.

SUPPLEMENTARY NOTE 2: SIMULATION OF DNSS FORMATION
We describe the nuclear spin dynamics in the pumpprobe experiments using the central spin box model.The Hamiltonian of the system reads where S is the hole spin, A is the joint hyperfine interaction constant for all N nuclei in the hole localization volume, I = N n=1 I n is the total nuclear spin composed of the individual spins I n .The hyperfine interaction is assumed here to be isotropic because of the dominant Stype Bloch wave functions at the top of the valence band in perovskites.The same description is valid for electrons in conventional GaAs-like semiconductors.However, in this work on perovskites the hyperfine interaction of the holes plays the main role.We note that the interaction with Br nuclei [S10] can reduce the efficiency of the spin transfer from holes to Pb, which can be absorbed in the renormalization of Γ 0 and ν 0 , see below.

A. PRC without DNSS
In equilibrium or at high enough polarization modulation frequencies, the nuclear spin distribution is Gaussian: with the parameter determining the dispersion of the distribution.In these conditions, the PRC is described by [S11] ( One can see from this expression that the HWHM of the PRC is given by ∆ B , which in combination with Supplementary Eq. ( 4) yields The constant A in the Hamiltonian, Supplementary Eq. ( 2), is related to the hyperfine coupling constant for a hole localized in a unit cell A 0 by A = A 0 /(N/β), where β = 0.22 is the natural abundance of 207 Pb.With this we arrive at which is used in the main text to determine the number of nuclei N .

B. Nuclear spin dynamics
To describe the nuclear spin dynamics, we note that in Supplementary Eq. ( 2) the total nuclear spin I is conserved.For N nuclear spins 1/2, the number of possible realizations of spin , where C k n is the binomial coefficient, and the probability to find the spin I is The eigenstates of the system can be labeled by the total angular momentum F z = I z + S z , and have the form where ↑ (↓) corresponds to a hole with spin S z = +1/2 (−1/2) and with ) and ) For a given I z after initialization of the hole in the spin-up state, the wave function is |I z , ↑ , which can be represented as where Then the coherence between the states Ψ + (I z + 1/2) and Ψ − (I z + 1/2) is lost [S12] on the timescale T * 2,h ∼ /(AI) T R , where T R is the repetition period of the pump pulses.As a result, the nuclear spin component I z increases by unity with the probability Here the argument I z + 1/2 is used to denote the total angular momentum component of the nuclei and the hole.
In the limit of classical I, this reduces to v( ).For a hole with spin down, the nuclear spin projection reduces by unity with the probability v(I z − 1/2).
In the limit of many nuclear spins, N 1, it is convenient to introduce the distribution function g(x), where x = I z /I, which is normalized such that It satisfies the Fokker-Planck equation where v = (1 − x 2 )/2.Introducing τ = t/(I 2 T R ) and Π(t) = P e (t)I, we obtain As boundary condition, g(x) should be finite at x = ±1.
From this description, one can see that the hole spin polarization is a function of the total nuclear spin I [see the definition below Eq. ( 1) and Supplementary Eq. ( 2)].So the correlations between distant nuclear spins also contribute to the observed Kerr rotation signal [S12-S14], which allows us to evidence nuclear spin squeezing and entanglement.

C. Nuclear spin inertia
To describe the nuclear spin inertia, we assume that each pump pulse excites a trion with the probability Γ 0 and after its recombination, the hole spin polarization increases by P h .According to the optical selection rules, the transverse hole spin components subsequently are erased [S15].This allows us to neglect the off-diagonal components of the hole and nuclei density matrix.In this case, the nuclear spin state is described by the probabilities P m for finding the spin I in the state with I z = m.We note that after the hole spin decoherence, the hole spin reduces by a factor of v(F z ), which provides an increase of I z .Thus, if the polarization modulation period consists of N mod pulses, the signal measured by the nuclear spin inertia method is given by [S16] where P m (k) describes the distribution of I z = m after k pulses, starting from the beginning of the period, and we assume that it is sufficient to consider only the first half of the period.This formalism allows us to simulate the evolution of the nuclear spin distribution function under pulsed excitation with modulation of the light helicity and to calculate the nuclear spin inertia signal as function of the magnetic field and the modulation frequency.Due to the numerical difficulties with the boundary conditions for Supplementary Eq. ( 18), we have used a finite number of nuclei, namely N = 60, as estimated in the main text.
To fit the frequency dependence of the PRC amplitude in Fig. 3(d), we have used Γ 0 = 0.003 and P h = 0.8.The latter parameter determines the saturation of the PRC amplitude at low modulation frequencies.The DNSS formation rate is given by it determines the characteristic time scale of the nuclear spin dynamics, or the rate of Overhauser field fluctuation rotation to the z axis.Thus, all parameters can be reliably determined from the experimental results.
The formation of the DNSS requires the rate ν 0 to be larger than the dephasing rate of the nuclei 1/T n 2 .It is determined by the local fields and can be estimated from the energy of the nuclear dipole-dipole interactions: where γ N is the lead gyromagnetic ratio and I n = 1/2 is the spin of 207 Pb nuclei.With the parameters given above (γ N /(2π) = 8.882 MHz/T, β = 0.221, a 0 = 0.6 nm) we obtain T n 2 ∼ 120 ms, which is indeed much larger than the DNSS formation time 1/ν 0 .So the nuclear spin dynamics is driven mainly by the hyperfine interaction, and the time scale of the appearance of the longitudinal nuclear spin polarization and suppression of the transverse nuclear spin fluctuations is much shorter than this estimation of T n 2 from the strength of the dipoledipole interaction.

D. Nuclear spin squeezing and entanglement
We characterize the shrinking of the nuclear spin distribution function along the transverse directions by the Kitagawa and Ueda spin squeezing parameter ξ s [S17, S18], which is defined by During the simulation of the nuclear spin inertia, it can be extracted as where the angular brackets denote the averaging over the total nuclear spin I with probabilities Supplementary Eq. ( 8), and P m symbolizes P m (k) with k = 1, for which the spin squeezing reaches its maximum.
The result of the calculation of the nuclear spin squeezing degree is shown in Fig. 3(e) by the solid line for the same parameters as in panel (d).To find ξ s from the experimental results, we assume that the nuclear spin dynamics obey the evolution according to Supplementary Eq. ( 2).Then, following the same procedure as described above, we find the DNSS that corresponds to the experimentally observed amplitude of the PRC at the given frequency and the calculated spin squeezing parameter for this state.The results are shown in Fig. 3(e) by the dots.
For low modulation frequencies we obtain ξ s = 0.48, which is limited by nuclear spin diffusion due to an incomplete hole spin polarization.In the limit of a perfect nuclear spin alignment along the z axis, the largest possible spin squeezing is ξ s = 0.44 because of the quantum fluctuations of the transverse components of the total nuclear spin I ∼ √ N ∼ 8.This situation can be called the Heisenberg limit, and it is only by 10% smaller than the experimentally reached value.
The large nuclear spin squeezing suggests deep entanglement between nuclear spins.One of the strict evidences of entanglement is the violation of the generalized spin squeezing inequality We note, that this represents a particular case of Eq. ( 2) with M = N , which means that there is at least any entanglement in the nuclear spins system.The PRC directly measures the transverse nuclear spin components I 2 x + I 2 y = I(I + 1) − I 2 z (from this relation one can see that I 2 x + I 2 y is smallest for the DNSS, where I z = I [S17]).To check the violation of Supplementary Eq. ( 24), we consider the upper limit for the longitudinal nuclear spin fluctuations (I z − I z ) 2 ≤ N/4, which corresponds to uncorrelated individual nuclear spins.From the definition in Supplementary Eq. ( 22) we obtain that states with ξ s ≤ 1/ √ 2 = 0.707 are entangled.This boundary is shown in Fig. 3(e) by the blue dashed line.
For the lowest polarization modulation frequency of f mod = 100 Hz, the PRC amplitude [L(B max ) − L(B = 0)]/L(B = 0) = 0.054 corresponds to I 2 x + I 2 y = 0.054I 2 , which after averaging over I yields I 2 x + I 2 y = 0.04N .Combined with the upper boundary for the longitudinal spin fluctuations we obtain which violates Supplementary Eq. ( 24) almost by a factor of two.In particular, using Eq. ( 2) we find that the achieved DNSS is at least equivalent to an M = 0. the lowest polarization modulation frequency the DNSS steady state is reached, where the depth of entanglement is limited by the conservation of I and by nuclear spin diffusion.For this state, the calculation of the nuclear spin fluctuations yields an M = 19-body entanglement.
We recall that the smaller M is, the deeper is the entanglement.

E. Comparison with the general central spin model
The box model has the advantage of simplicity, however it can underestimate the nuclear spin polarization degree [S19, S20].In order to compare the general central spin model with the box model, we consider in this subsection N = 10 nuclear spins in the vicinity of a hole bound by a zero range potential.The distribution of the hyperfine coupling constants is exponential in this case and we take its average to be equal to the parameter A in the Hamiltonian of the box model.Then we consider hole spin orientation by the train of circularly polarized pulses with the period τ .The hole spin polarization before the next pulse for Aτ / 1 is limited by the hole spin dephasing in the nuclear field.We show it as a function of the number of applied pulses in Supplementary Fig. 5.For many pulses, the spin polarization tends to 1, which is analogous to the disappearance of the dip in the PRC.One can see that this dependence is similar for the homogeneous and inhomogeneous hyperfine interaction.
Let us explain the similarity of the results obtained in the two models.(i) In the case of the box model, the nuclear spin polarization saturates at a small value ∼ 1/ √ N , in this case the fluctuations of the total nuclear spin are all rotated towards the z axis.In order to rotate a typical fluctuation of I ∼ √ N one needs ∼ √ N pulses (for Aτ / 1).Afterwards, the nuclei are in the DNSS.At the same time, the hole spin, which is oriented along this axis by every optical pulse, does not precess, so there is no dephasing.(ii) For the inhomogeneous hyperfine interaction, a similar number of pulses √ N rotates the fluctuations of the Overhauser field towards the z axis, which leads to the formation of the DNSS and suppression of the transverse fluctuations of the Overhauser field.However, afterwards the nuclear spin polarization continues to increase due to the slow destruction of the DNSS [S20].After a large number of pulses, the spin polarization can be large, of the order of 100% in contrast to the box model.Accordingly, the black curve in Supplementary Fig. 5 increases for N ∼ 3 as fast as the red one, but then it saturates slower than the red one.
In the nuclear spin inertia measurements, we observe the suppression of the hole spin dephasing and the absence of significant nuclear spin polarization.This corresponds to time scales, when the fluctuations of the Overhauser field are rotated towards the optical axis, but a significant nuclear spin polarization is not reached.So the box model and the general central spin model give similar results on this time scale, which demonstrates applicability of the box model to the description of the nuclear spin squeezing and formation of the DNSS.
(a) Photoluminescence spectrum of the FAPbBr3 crystal measured under continuous-wave excitation with the photon energy of 3.06 eV at T = 5 K. Inset shows a photo of the studied crystal.(b) Example of the TRKE dynamics (black line) at the pump-probe energy of 2.191 eV.The pump beam was helicity modulated between σ + /σ − circular polarization at the frequency of 50 kHz.Its fit with Supplementary Eq. (1) is shown by the yellow dashed line.The electron and hole spin precession components evaluated from the fit are shown below by the red and blue lines, respectively.
Supplementary Figure 2. (a,b) Overhauser field (B N,e(h) ) experienced by the hole and electron spins as function of the magnetic field tilted by θ = 30 Supplementary Figure 3. (a) PRC with constant pump helicity σ + (blue) or σ − (red).T = 1.6 K, pump power 6 mW.(b) Sum of the two signals from panel (a).(c) Difference of the two signals from panel (a) (green), corresponding to the nuclear spin inertia signal at f mod = 0, and nuclear spin inertia signal at f mod = 20 Hz (magenta).(d) Scheme of the applied excitation.at different pump powers P , for (a) f mod = 20 Hz and (b) f mod = 5 kHz.The curves are not shifted.(c,d,e) Parameters evaluated from curves (a,b), full symbols for f mod = 5 kHz and open symbols for f mod = 20 Hz.(c) Pump power dependence of the amplitudes of the broad A b and narrow An dips (red & dark red diamonds and green triangles, respectively); high field (150 mT) values of the amplitudes A hf (blue & light blue squares).(d) Pump power dependence of the ratio of amplitude of the broad dip and its high field value (A b /A hf ) for f mod = 20 Hz (red) and f mod = 5 kHz (blue).(e) Pump power dependence of the HWHM of the broad (open orange diamonds) and narrow (open green triangles) PRC dips at f mod = 20 Hz, and of the broad dip at f mod = 5 kHz (filled orange diamonds).For all panels T = 1.6 K.
Supplementary Figure5.Hole spin polarization before the next pump pulse arrival as a function of the number of circularly polarized pulses calculated for the inhomogeneous (black solid curve) and homogeneous (red dashed curve) hyperfine interaction for 10 nuclear spins and τ A/ = 17 (exact value of this parameter is not important as long as it is large).